Model Checking Almost All Paths Can Be Less Expensive Than Checking All Paths

  • Matthias Schmalz
  • Hagen Völzer
  • Daniele Varacca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4855)


We compare the complexities of the following two model checking problems: checking whether a linear-time formula is satisfied by all paths (which we call universal model checking) and checking whether a formula is satisfied by almost all paths (which we call fair model checking here). For many interesting classes of linear-time formulas, both problems have the same complexity: for instance, they are PSPACE-complete for LTL.

In this paper, we show that fair model checking can have lower complexity than universal model checking, viz., we prove that fair model checking for L(F ∞ ) can be done in time linear in the size of the formula and of the system, while it is known that universal model checking for L(F ∞ ) is co-NP-complete. L(F ∞ ) denotes the class of LTL formulas in which F ∞  is the only temporal operator. We also present other new results on the complexity of fair and universal model checking. In particular, we prove that fair model checking for RLTL is co-NP-complete.


Model Check Linear Time Temporal Logic Universal Model Linear Temporal Logic 


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  1. 1.
    Alur, R., Henzinger, T.A.: Local liveness for compositional modeling of fair reactive systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 166–179. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  2. 2.
    Berwanger, D., Grädel, E., Kreutzer, S.: Once upon a time in the west - determinacy, definability, and complexity of path games. In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 229–243. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Emerson, E.A.: Temporal and modal logic. Handbook of Theoretical Computer Science B(16), 995–1072 (1990)MathSciNetGoogle Scholar
  5. 5.
    Emerson, E.A., Lei, C.-L.: Modalities for model checking: Branching time logic strikes back. Sci. Comput. Program. 8(3), 275–306 (1987)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Lichtenstein, O., Pnueli, A.: Checking that finite state concurrent programs satisfy their linear specification. In: POPL, pp. 97–107. ACM Press, New York (1985)Google Scholar
  7. 7.
    Schmalz, M.: Extensions of an algorithm for generalised fair model checking. Diploma Thesis, Technical Report B 07-01, University of Lübeck, Germany (2007),
  8. 8.
    Schmalz, M., Völzer, H., Varacca, D.: Model checking almost all paths can be less expensive than checking all paths. Technical Report 573, ETH Zürich, Switzerland (2007),
  9. 9.
    Schnoebelen, P.: The complexity of temporal logic model checking. In: AiML, pp. 393–436. King’s College Publications (2002)Google Scholar
  10. 10.
    Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. J. ACM 32(3), 733–749 (1985)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Sistla, A.P., Vardi, M.Y., Wolper, P.: The complementation problem for Büchi automata with applications to temporal logic. In: Brauer, W. (ed.) Automata, Languages and Programming. LNCS, vol. 194, pp. 465–474. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  12. 12.
    Varacca, D., Völzer, H.: Temporal logics and model checking for fairly correct systems. In: LICS, pp. 389–398. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  13. 13.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: FOCS, pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)Google Scholar
  14. 14.
    Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: LICS, pp. 332–344. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  15. 15.
    Völzer, H., Varacca, D., Kindler, E.: Defining fairness. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 458–472. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Zuck, L.D., Pnueli, A., Kesten, Y.: Automatic verification of probabilistic free choice. In: Cortesi, A. (ed.) VMCAI 2002. LNCS, vol. 2294, pp. 208–224. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Matthias Schmalz
    • 1
  • Hagen Völzer
    • 2
  • Daniele Varacca
    • 3
  1. 1.ETH ZürichSwitzerland
  2. 2.IBM Zurich Research LaboratorySwitzerland
  3. 3.PPS - CNRS & Univ. Paris 7France

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